COMPUTATION OF π_{∗}(Q(2) ∧ V (0))

MARK BEHRENS

In this section we will run the v_{1} Bockstein spectral sequence (v_{1}-BSS) on the computations of Section ?? to produce a computation of π_{∗}(Q(2) ∧ V (0)) at the prime 3. We shall then compare this to the computations of Shimomura [?], and deduce the homotopy of Q′(2) ∧ V (0). It is actually easier to run the v_{1}-BSS on the E_{2}-term of the ANSS for Q(2)∧V (1), computing the ANSS E_{2} term for Q(2)∧V (0). The homotopy is then obtained by the differentials forced on the latter from the ANSS differentials for T MF.

1. The v_{1}-BSS for E_{2}(Q(2) ∧ V (0))

The computations in this section use the same methods that Miller, Ravenel, and Wilson originally used to compute the 2-line of the ANSS at odd primes [?]. Recall that v_{1} is represented by b_{2} and q_{2} in the rings B and MF (2)[1/2], respectively.

We summarize the computation of the v_{1}-BSS

E_{2}(Q(2) ∧ V (1)) ⊗ P [v_{1}] ⇒ E_{2}(Q(2) ∧ V (0)).

We first compute the BSS differentials originating from the elements v_{2}^{k} ∈ E_{2}^{0,∗}(Q(2)∧ V (1)) (Adams-Novikov filtration 0).

Define elements y_{i}, for i ≥ 2 and x_{i} for i ≥ 0 in C^{0}(Q(2)) by

y_{i} = b_{4}^{3^{i}+3^{i−1}}b_{2}^{3^{i}+3^{i−1}−3}b_{6} − b_{4}^{3^{i}+3^{i−1}+1}b_{2}^{3^{i}+3^{i−1}−2} − b_{4}^{3^{i}+3^{i−1}+2}b_{2}^{3^{i}+3^{i−1}−4}r

x_{0} = b_{4}^{2}
x_{1} = x_{0}^{3} + b_{4}^{3}b_{2}^{3}b_{6}
x_{2} = x_{1}^{3} + y_{2} + b_{4}^{11}b_{2}^{11}b_{6}
x_{i} = x_{i−1}^{3} + y_{i} for i ≥ 3.

The elements x_{i} have the property that

x_{i} ≡ b_{4}^{2·3^{i}} (mod b_{2}).

When one computes their differentials in the complex C^{∗}(Q(2)), one finds

d(x_{0}) = −b_{2}b_{4}r
d(x_{i}) ≡ −b_{4}^{3^{i}+3^{i−1}}b_{2}^{3^{i}+3^{i−1}−1}r (mod v_{1}^{3^{i}+3^{i−1}}) for i ≥ q

One concludes that there are v_{1}-BSS differentials (for s ≢ 0 (mod 3))

d_{1}(v_{2}^{s}) = −sv_{1}v_{2}^{s−1}h_{1}
d_{3^{i}+3^{i−1}−1}(v_{2}^{3^{i}s}) = −sv_{1}^{3^{i}+3^{i−1}−1}v_{2}^{(3^{i}(s−1)+3^{i−1})/2}α for i ≥ 1.

The elements on the 1-line E_{2}^{1,∗}(Q(2) ∧ V (1)) are of the form v_{2}^{k}h_{1}, v_{2}^{k}α and v_{2}^{k}ζ. Of these, a basis of the elements that were the targets of v_{1}-BSS differentials is given by

v_{2}^{k}h_{1} k ≡ 0, 1 (mod 3)
v_{2}^{k}α k ≡ 2 · 3^{i} (mod 3^{i+2})
v_{2}^{k}α k ≡ 2 · 3^{i} + 2 · 3^{i+1} (mod 3^{i+2}).

Of the elements on the 1-line which are not of the form ζv_{2}^{k}, the only one that support non-trivial v_{1}-BSS differentials are of the form

d_{2}(v_{2}^{k}α) = ±v_{1}^{2}v_{2}^{k−1}β k ≡ 1 (mod 3)
d_{2}(v_{2}^{k}h_{1}) = ±v_{1}^{2}v_{2}^{k−2}b_{1} k ≡ 2 (mod 3)

That leaves elements of the form

v_{2}^{k}α k ≡ 3^{i} (mod 3^{i+1}), i ≥ 1

to support infinite v_{1}-towers.

Finally, of the elements v_{2}^{k}b_{1}, we have v_{1}-BSS differentials

d_{1}(v_{2}^{k}b_{1}) = ±v_{2}^{k+1}βα k ≢ 0 (mod 3).

The d_{1} and d_{2} differentials are propagated by β multiplication, and all of the differentials are propagated by ζ multiplication.

We are left with the following description of E_{2}(Q(2) ∧ V (0)). By a β − α-tower, we mean a free module over P [β_{1}]⊗E[α_{1}]. When we refer to a reduced β −α tower, we mean the ideal of P [β_{1}] ⊗ [α_{1}] generated by α_{1}.

Proposition 1.1. We have the following description of E_{2}(Q(2) ∧ V (1)). In all that follows, k may be any integer, s any integer which 3 does not divide, and i any non-negative integer. Define

a_{i} = \begin{cases}1, & i = 0 \\ 3^{i} + 3^{i−1} − 1, & i > 0. \end{cases}

Then the E_{2}-term has following generators with the indicated v_{1}-orders.

1(∞), β̃_{3^{i}s/a_{i}}, α_{1}(∞), v_{2}^{3^{i+1}+3^{i+2}s}α_{1}(∞)

Except for 1, all of the above generators live on the 1-line. The elements 1, β̃_{(1+3k)/1}, and β̃_{3s/3} all support β − α towers. The elements β̃_{(2+3k)/1}, β̃_{3^{i+2}s/a_{i+2}}, and v_{2}^{3^{i+1}+3^{i+2}s}α_{1} support reduced β − α towers. The E_{2}-term is all of the above tensored with E[ζ].

The differentials of the ANSS are easily deduced from the differentials in the ANSS for T MF.

2. Description of E_{2}(Q′(2) ∧ V (0))

The E_{2} term E_{2}(Q′(2) ∧ V (0)) is characterized by having the following v_{1}-BSS differentials

d_{4·3^{n}}(v_{2}^{k}ξ) = v_{1}^{4·3^{n}}v_{2}^{l}ψ_{1}

where l is appropriate for k and n, and k satisfies

k ≡ 1 + 3 + 3^{2} + ⋯ + 3^{n−1} + 2 · 3^{n} (mod 3^{n+2})

COMPUTATION OF π_{∗}(Q(2) ∧ V (0)) 3

or

k ≡ 1 + 3 + 3^{2} + ⋯ + 3^{n−1} + 2 · 3^{n} + 2 · 3^{n+1} (mod 3^{n+2}).

Every v_{2} multiple v_{2}^{l}ψ_{1} has a finite v_{2}-tower created by these BSS differentials The elements v_{2}^{k}ξ for k not of one of the forms above support infinite v_{1} towers. In L_{K(2)}V (0), these must be turned into v_{1}-torsion with the help of the infinite v_{1}-towers supported by the elements of the form

v_{2}^{3^{i+1}+3^{i+2}s}α_{1}

in E_{2}(Q(2) ∧ V (0)).

This provides a computational test for any putative description of the attaching map of Q(2) to Q′(2).